Theorem spw 2037

Description: Weak version of the specialization scheme sp 2176. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 2176 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 2176 having mutually distinct setvar variables and no wff metavariables (see ax12wdemo 2131 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 2176 are spfw 2036 (minimal distinct variable requirements), spnfw 1983 (when 𝑥 is not free in ¬ 𝜑), spvw 1984 (when 𝑥 does not appear in 𝜑), sptruw 1809 (when 𝜑 is true), spfalw 2001 (when 𝜑 is false), and spvv 2000 (where 𝜑 is changed into 𝜓). (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 27-Feb-2018.)

Hypothesis

Ref	Expression
spw.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
spw	⊢ (∀𝑥𝜑 → 𝜑)

Distinct variable groups:   𝑥,𝑦   𝜓,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem spw

Step	Hyp	Ref	Expression
1		ax-5 1913	. 2 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓)
2		ax-5 1913	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)
3		ax-5 1913	. 2 ⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑)
4		spw.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	1, 2, 3, 4	spfw 2036	1 ⊢ (∀𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783

This theorem is referenced by:  hba1w  2050  19.8aw  2053  exexw  2054  spaev  2055  ax12w  2129  nfcriOLD  2897  rspw  3130  reldisj  4385  ralidmw  4438  dtruALT2  5293  dtru  5359  bj-ssblem1  34835  bj-ax12w  34858